Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$, where $\lambda > \mu$ and let $c>0$. I need to find the conditional density function of $X$ given that $X + Y=c$. Here is what I have $$f(X=x|X+Y=c)=\frac{f(X=x,X+Y=c)}{f(X+Y=c)}$$ $$=\frac{f(X=x,Y=c-x)}{f(X+Y=c)}$$ $$=\frac{f(X=x)f(Y=c-x)}{f(X+Y=c)}$$
I got the last step by knowing $X$ and $Y$ are independent. I'm a little unsure that the second step I did is valid where I changed $X+Y=c$ into $Y=c-x$. If those parts are both correct, my only issue is the pdf of $X+Y$. I think that it is Gamma, but I'm not sure about the parameterization.